Energy Band View of Semiconductors Conductors, semiconductors, insulators: Why is it that when individual atoms get close together to form a solid – such as copper, silicon, or quartz – they form materials that have a high, variable, or low ability to conduct current? Understand in terms of allowed, empty, and occupied electronic energy levels and electronic energy bands. Fig. 1 shows the calculated allowed energy levels for electrons (vertical axis) versus distance between atoms (horizontal axis) for materials like silicon.

Fig. 1. Calculated energy levels in the diamond structure as a function of assumed atomic spacing at T = 0 o K. (From “Introduction to Semiconductor Physics”, Wiley, 1964)

In Fig. 1, at right atoms are essentially isolated; at left atomic separations are just a few tenths of a nanometer, characteristic of atoms in a silicon crystal. If we start with N atoms of silicon at the right, which have 14 electrons each, there must be 14N allowed energy levels for the electrons. (You learned about this in physics in connection with the Bohr atom, the Pauli Exclusion principle, etc.) If the atoms are pushed together to form a solid chunk of silicon, the electrons of neighboring atoms will interact and the allowed energy levels will broaden into energy bands.

When the “actual spacing” is reached, the quantum-mechanical calculation results are that: at lowest energies very narrow ranges of energy are allowed for inner electrons (these are core electrons, near the nuclei); a higher band of 4N allowed states exists that, at 0 o K, is filled with 4N electrons; then an energy gap, E G, appears with no allowed states (no electrons permitted!); and at highest energies a band of allowed states appears that is entirely empty at 0 o K. Can this crystal conduct electricity?

NO, it cannot conductor electricity at 0 o K because that involves moving charges and therefore an increase of electron energy – but we have only two bands of states separated by a forbidden energy gap, E G. The (lower) valence band is entirely filled, and the (upper) conduction band states are entirely empty. To conduct electricity we need to have a band that has some filled states (some electrons!) and some empty states that can be occupied by electrons whose energies increase.

Fig. 2 shows the situation at 0 o K for (left) a metallic solid such as copper, and (right) a semiconductor such as silicon. The metal can conduct at 0 o K because the uppermost band contains some electrons and some empty available energy states. The semiconductor cannot conduct – it is an insulator. If we raise the temperature of the semiconductor, some electrons in the filled valence band may pick up enough energy to jump up into an unoccupied state in the conduction band. Thus, at a finite temperature, a pure (intrinsic) semiconductor has a finite electrical conductivity.

Fig. 2. Electronic energy bands for (a) metallic conductor at T = 0 o K; (b) insulator or intrinsic semiconductor at 0 o K.

How much conductivity can a pure (intrinsic) semiconductor exhibit? This depends on how much thermal energy there is and the size of the energy gap, E G : Mean thermal energy is kT, where k = Boltzmann’s constant = 1.38 x J/K and T is the absolute temperature. In electron volts this is kT/q e, or 26 millivolts for room temperature (300 o K) For silicon, E G = 1.12 eV at 300 o K This leads in pure (intrinsic) Si to a carrier concentration n i = carriers/cm 3 at 300 o K

Adding Impurities (Doping) to Adjust Carrier Concentrations Adjust carrier concentrations locally in semiconductor by adding easily ionized impurities to produce mobile electrons and/or holes To make silicon N-type: Add valence 5 phosphorous (P) atoms to valence 4 silicon. Fifth electron is easily freed from the atom by a little thermal energy (0.045 eV for phosphorous) to create (donate) a mobile electron. Fig. 3a shows the donor energy level just below bottom of conduction band. To make silicon P-type: Add valence 3 boron (B) to silicon. An electron at the top of the valence band can pick up enough thermal energy to release it from the silicon so it attaches to a boron atom, completing its outer ring of electrons. In the band picture, Fig. 3b, this is represented by an acceptor. energy level eV above the top of the valence band.

Fig. 3a. Electronic energy band for n-type semiconductor (Ge) with donors only.

Fig. 3b. Electronic energy band for p-type semiconductor (Ge) with acceptors only.